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Chapter 3 Solutions
Elements Of Modern Algebra
- If a is an element of order m in a group G and ak=e, prove that m divides k.arrow_forward42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .arrow_forward43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .arrow_forward
- (See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup that leaves each of the elements 1,2,...,i fixed: Ki=gGg(k)=kfork=1,2,...,i For i=1,2,...,n. Prove that G=Sn if and only if HiHj for all pairs i,j such that ij and in1. A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.arrow_forwardFind a subset of Z that is closed under addition but is not subgroup of the additive group Z.arrow_forward15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.arrow_forward
- Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.arrow_forwardLabel each of the following statements as either true or false. The Generalized Associative Law applies to any group, no matter what the group operation is.arrow_forwardLet be a subgroup of a group with . Prove that if and only ifarrow_forward
- Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.arrow_forwardExercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.arrow_forward39. Assume that and are subgroups of the abelian group. Prove that the set of products is a subgroup of.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,